# Block matrix with Lower triangular matrices as blocks

Consider a block matrix A

$$A = \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{pmatrix}_{2 \times 2}$$

such that its entries $A_{ij}$ are lower triangular matrices. What is the inverse of A? How and which norm can be used to show that the norm of matrix A is less than or equal one? And is there any relation between the eigenvalues of $A$ and the eigenvalues of lower triangular matrices $A_{ij}$?

if $A = \left [ \matrix{X & Y \\ W & Z} \right ]$, and we use the norm $\|M\| = \sqrt{\text{tr}(M^*M)}$, for all of them, then it is easy to see that $$\|A\|^2 = \|X\|^2 + \|Y\|^2 +\|W\|^2 +\|Z\|^2$$